Problem: Simplify the following expression: $k = \dfrac{-108r^2 + 24r}{-36r^2 + 48r}$ You can assume $r \neq 0$.
Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-108r^2 + 24r = - (2\cdot2\cdot3\cdot3\cdot3 \cdot r \cdot r) + (2\cdot2\cdot2\cdot3 \cdot r)$ The denominator can be factored: $-36r^2 + 48r = - (2\cdot2\cdot3\cdot3 \cdot r \cdot r) + (2\cdot2\cdot2\cdot2\cdot3 \cdot r)$ The greatest common factor of all the terms is $12r$ Factoring out $12r$ gives us: $k = \dfrac{(12r)(-9r + 2)}{(12r)(-3r + 4)}$ Dividing both the numerator and denominator by $12r$ gives: $k = \dfrac{-9r + 2}{-3r + 4}$